The aggregation equation with power-law kernels: ill-posedness, mass concentration and similarity solutions

We study the multidimensional aggregation equation u_t+\Div(uv)=0, $v=-\nabla K*u$ with initial data in \cP_2(\bR^d)\cap L_{p}(\bR^d). We prove that with biological relevant potential $K(x)=|x|$, the equation is ill-posed in the critical Lebesgue space L_{d/(d-1)}(\bR^d) in the sense that there exists initial data in \cP_2(\bR^d)\cap L_{d/(d-1)}(\bR^d) such that the unique measure-valued solution leaves L_{d/(d-1)}(\bR^d) immediately. We also extend this result to more general power-law kernels $K(x)=|x|^\alpha$, $0<\alpha<2$ for $p=p_s:=d/(d+\alpha-2)$, and prove a conjecture in Bertozzi, Laurent and Rosado [5] about instantaneous mass concentration for initial data in \cP_2(\bR^d)\cap L_{p}(\bR^d) with $p<p_s$. Finally, we classify all the "first kind" radially symmetric similarity solutions in dimension greater than two.Comment: typos corrected, 18 pages, to appear in Comm. Math. Phy

Solvability of second-order equations with hierarchically partially BMO coefficients

By using some recent results for divergence form equations, we study the $L_p$-solvability of second-order elliptic and parabolic equations in nondivergence form for any $p\in (1,\infty)$. The leading coefficients are assumed to be in locally BMO spaces with suitably small BMO seminorms. We not only extend several previous results by Krylov and Kim [14]-[18] to the full range of $p$, but also deal with equations with more general coefficients.Comment: 28 Pages. An earlier version was submitted in 2009. The current version is to appear in Trans. Amer. Math. So